# Dispersion and Stability Condition of Seismic Wave Simulation in TTI Media

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## Abstract

For seismic waveform simulation in tilted transversely isotropic (TTI) media, we derive explicitly the numerical dispersion relation and the stability condition for the computation of a 2D pseudo-acoustic wave equation. The numerical dispersion relation indicates that the number of sampling points per wavelength has the greatest influence on the dispersion, while the anisotropic parameters of the TTI media and the mesh rotation angle have little influence on the dispersion. Given an appropriate spatial sampling, the stability condition is for the selection of the time step for the implementation of the TTI wave equation. We partition a numerical model using quadrangle grids in Cartesian coordinates, and map it to a computing model in which any non-rectangular meshes in Cartesian coordinates become rectangular meshes. Then we reformulate the pseudo-acoustic wave equation for the TTI media accordingly in the computational space. We implement seismic waveform simulation using the second-order finite-difference method straightforwardly, and show examples with a desirable accuracy using a model with non-rectangular meshes in Cartesian coordinates along a curved surface and fluctuating interfaces in the TTI media.

## Keywords

Anisotropy dispersion finite difference stability TTI wave equation## 1 Introduction

Seismic anisotropy commonly exists in the Earth’s subsurface media (Tsvankin et al. 2010; Takanashi and Tsvankin 2012). Accurate waveform simulation in tilted transversely isotropic (TTI) media is of importance for seismic waveform inversion. The latter reconstructs the subsurface velocity model quantitatively based on seismic waveform data. Seismic waveform data routinely recorded by hydrocarbon explorations comprise of mainly P-wave reflection data. Therefore, we use an acoustic wave equation in this paper for waveform simulation through TTI media containing a curved surface and fluctuating interfaces. This waveform simulation scheme is a core engine employed by the iterative inversion of seismic P-wave reflection data (Wang 2003; Wang and Rao 2009).

Although the P-wave and S-wave are coupled in the elastic wave equation in TTI media, we can have a pseudo-acoustic wave equation if the S-wave velocity is fixed along the axis of symmetry (Alkhalifah 1998; Fletcher et al. 2009). This acoustic wave equation is defined by two anisotropic parameters \(\varepsilon\) and *δ* measuring the difference between two axes of the elliptic wavefront and the deviation from a perfect elliptical shape, respectively (Thomsen 1986). In the context of seismic waveform inversion, Pratt and Shipp (1999) and Rao and Wang (2011) use an acoustic wave equation defined by a single anisotropic parameter \(\varepsilon\). Rao et al. (2016) provide the derivation of this wave equation with a single anisotropic parameter. In this paper, we adopt the acoustic wave equation with two anisotropic parameters. That is the pseudo-acoustic wave equation.

We also consider the models with a curved surface and fluctuating interfaces in the subsurface. We use quadrangle grids to partition every individual layer, confined by two fluctuating interfaces, based on a body-fitting scheme. By solving Poisson’s equation, these grids also satisfy the pseudo-orthogonal condition (Rao and Wang 2013) in which grids should have the acute angles > 67° (90° for completely orthogonal grids). We transform non-rectangular meshes in Cartesian coordinates into rectangular ones through conformal mapping. These grids, as structured, keep the similar neighborhood relationships as they do in Cartesian coordinates. We reformulate the pseudo-acoustic wave equation accordingly in the computational space. For the reformulated TTI wave equation, we analytically derive the corresponding numerical dispersion relation and stability condition, which provide the basis of finite-difference parameter selection of the TTI media waveform simulation.

## 2 Wave Equation

*x*,

*z*) are Cartesian coordinates. Note that we use \(H_{x}\) and \(H_{z}\) here to present differentials with respect to \(\hat{x}\) and \(\hat{z}\), respectively. Fletcher et al. (2009) used \(H_{1}\) and \(H_{2}\) to present the two differentials with respect to \(\hat{z}\) and \(\hat{x}\), respectively. For the completeness, we summarize the derivation of this pseudo-acoustic wave equation in the appendix.

In the pseudo-acoustic wave equation [Eq. (1)], the two anisotropic velocities (\(v_{px}\), \(v_{pn}\)) are related to the TTI anisotropy parameters by \(v_{px} = v_{pz} \sqrt {1 + 2\varepsilon }\) and \(v_{pn} = v_{pz} \sqrt {1 + 2\delta }\), where \(\varepsilon\) and \(\delta\) are two anisotropy parameters (Thomsen, 1986). Note that the SV-wave velocity \(v_{sz}\) in Eq. (1) does not have a significant effect on the P wavefront. It would have an effect on the SV-wave wavefront and in turn on the P–S converted wave imaging. However, from the P–P wave imaging/gradient calculation point of view, the SV-wave wavefront (traveling slowly) is simply an unwanted artefact and does not significantly affect the P-wave image.

*x*,

*z*) to the computational space (\(\xi ,\eta\)). In the computational space, the first-order spatial derivatives become

Substituting these spatial derivatives into the 2D differential operators \(H_{x}\) and \(H_{z}\), we rewrite wave equation [Eq. (1)] in the computational space (\(\xi , \, \eta\)). We numerically solve this time-space domain wave equation in the computational space using a second-order finite-difference method.

## 3 Numerical Dispersion

We analyze the numerical dispersion in this section, for the finite-difference implementation of the spatial derivatives in the pseudo-acoustic wave equation.

*s*, which is the reciprocal number of sampling points per wavelength. In this test, the Courant number \(Q = 0.1\), the anisotropy parameter \(\varepsilon = 0.1\), the angle between the symmetry axis and the vertical direction of the TTI medium is \(\phi = 30^{ \circ } ,\) and the meshing rotation angle \(\varphi = 30^{ \circ } .\)

Numerical tests indicate that the dispersion is weakest when the propagation direction is perpendicular to the axis of TTI media (\(\theta = 90^{ \circ }\)). The dispersion is slightly increased when the propagation direction is close to the axis of symmetry (\(\theta \to 0^{ \circ }\)).

As it is well known, a general requirement for forward modeling in the isotropic media with the second-order finite-difference simulations is 10 grids per wavelength (Kreiss and Oliger 1972; Wu et al. 1996). Figure 3 indicates that satisfying the condition \(s \le 0.1\), that is \(\ge\) 10 grids per wavelength, means that the dispersion is less than 1% in wavefield simulation through anisotropic media. The smallest dispersion ratio is 0.99, for the anisotropic parameters \(\varepsilon\) and \(\delta\) between \(-0.25\) and \(0.25\) and for the mesh rotation angle \(\varphi\) between 0 and 90^{o}. These numerical tests confirm that, as long as \(s = 0.1\), the dispersion requirement of forward modeling is still satisfied.

Note that the points per wavelength quoted above is for wave propagation within one wavelength. Because of the error accumulation, the number of points required for a given tolerance depends on the number of wavelengths the wave is to propagate. According to Kreiss and Petersson (2012), a typical scaling is \((\lambda /\mu )^{1/n}\), where (\(\lambda\), \(\mu\)) here are the Lame parameters, and \(n\) is the order of accuracy of the method.

## 4 Numerical Stability

We derive the stability condition in this section, for the numerical implementation of the time derivatives in the wave equation, given appropriate sampling in the spatial domain. We investigate this numerical stability when the spatial grids are discretized in Cartesian coordinates.

Expression (34) is the stability condition for the \(O(\Delta t^{2} , \, h^{2} )\) scheme, the second-order finite-differencing in both time and space. For selecting the time step using this expression, we set \(h\) as the smallest cell size of non-rectangular grids generated by body fitting. It is worth mentioning that, setting \(\varepsilon = 0\), the stability condition will reduce to the well-known formula of an \(O(\Delta t^{2} , \, h^{2} )\) scheme in isotropic media (Lines et al. 1999).

## 5 Waveform Simulation in TTI Media

Figure 4a shows that the body-fitted grids coincide well with the curved surface and a fluctuating interface at the middle of the model, and, meanwhile, there are non-rectangular meshes unavoidably existing on either side of the interface. As long as the media are discretized by body-fitted grids, there is no explicit enforcement of the normal stress and displacement continuity conditions at the interface. In order to avoid any instability caused by the low meshing precision, we adapt a summation-by-parts (SBP) finite-difference method (Kreiss and Scherer 1974; Nilsson et al. 2007) to the case here with the cell size variation of body-fitted grids. We use the finite-difference operators with the second-order accuracy in both temporal and spatial directions. According to Sjögreen and Petersson (2012), the SBP method is automatically stable for partial differential equations with the second-order spatial derivatives. The pseudo-acoustic wave equation is a system of partial differential equations with an initial condition which defines the source signature. For the absorbing boundary, we use the perfectly matched layer method, presented in the computational space, and discretized by finite differencing with a second-order accuracy (Rao and Wang 2013).

The snapshots (Fig. 4b, c) indicate that the non-rectangularity in the mesh does not have visible effect deteriorating wavefield simulation, once the numerical dispersion relation and the stability condition are satisfied. This example demonstrates that this numerical simulation method for a model with such irregular grids is able to produce an accurate wavefield, without any artificial reflections from the interface. In contrast, if using a standard finite-difference method, there must be some artificial reflections caused by strong variation in the cell sizes of body-fitted grids, and even the two-layer velocities are assumed to be constant.

Note that the geometrical configuration of Figs. 4a and 6a is the same. In waveform simulation, we always implement two steps. In the first step, we use a model with a constant velocity for all layers, such as Fig. 4a, to check the effectiveness of grids. In the second step, we use the proper model with layered velocities, such as Fig. 6a, to generate the desired waveform.

## 6 Conclusions

For seismic waveform simulation with the TTI wave equation, we have derived a numerical dispersion relation, and demonstrated that the number of sampling points per wavelength has the greatest influence on the dispersion, and both the anisotropic parameters and the mesh rotation angle have little influence on the dispersion. Further, we have derived the condition for the numerical stability for the selection of time step, given the spatial sampling is appropriately selected in Cartesian coordinates. In the derivation, we have used the second-order finite-differencing operators in both time and space and assumed an elliptical anisotropy.

## Notes

### Acknowledgements

The authors are grateful to the National Natural Science Foundation of China (grant no. 41622405), the Science Foundation of China University of Petroleum (Beijing) (grant no. 2462018BJC001), and the sponsors of the Centre for Reservoir Geophysics, Imperial College London, for supporting this research.

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